Volume 90 BULLETIN of The October 2020 INSTITUTE of COMBINATORICS and its APPLICATIONS Editors-in-Chief: Marco Buratti, Donald Kreher, Ortrud Oellermann, Tran van Trung ISSN: 2689-0674 (Online) Boca Raton, FL, U.S.A. ISSN: 1183-1278 (Print) BULLETIN OF THE ICA Volume 90 (2020), 107{138 Small unit-distance graphs in the plane 1 2 Aidan Globus and Hans Parshall∗ 1 The Ohio State University, Columbus, Ohio, U.S.A. [email protected] 2 Western Washington University, Bellingham, Washington, U.S.A. [email protected] Abstract We prove that a graph on up to 9 vertices is a unit-distance graph if and only if it does not contain one of 74 so-called minimal forbidden graphs. This extends the work of Chilakamarri and Mahoney (1995), who provide a similar classification for unit-distance graphs on up to 7 vertices. 1 Introduction A graph G on vertices V is said to be unit-distance if there exists an embedding ϕ: V R2 such that for every pair of adjacent vertices v, w V , we have ϕ(v) ϕ→(w) = 1; here and throughout we use to denote∈ the usual Euclidean| − norm.| Several long-standing open problems| · | concern unit- distance graphs. For instance, Erd˝os’unit-distance problem asks for the maximum number of edges u(n) over all unit-distance graphs on n vertices. 1+ c The lower bound of u(n) n log log n for some fixed c > 0 has not been improved since the 1946 paper≥ of Erd˝os[6], and it is suspected to be close to the truth. Spencer, Szemer´ediand Trotter [13] established the upper bound ∗Corresponding author. Key words and phrases: unit-distance graphs, graph embeddings AMS (MOS) Subject Classifications: 52C10, 05C35 Received: 7 September 2019 107 Accepted: 5 April 2020 Globus and Parshall Figure 1: We will see that one of these two graphs is unit-distance (see Figure5), while the other is forbidden (see Lemma 32). of u(n) Cn4/3 for some fixed C > 0. While this has not been improved, there are≤ now several distinct proofs of the same upper bound; see [14] for a particularly elegant argument and a more thorough overview. Erd˝osalso popularized the Hadwiger–Nelson problem, which asks for the chromatic number χ(R2) of the infinite unit-distance graph with vertex set R2 and an edge between v, w R2 exactly when v w = 1. The history of the problem is carefully documented∈ by Soifer| [12−]; he| credits Nelson with the lower bound of χ(R2) 4 and Isbell with the upper bound of χ(R2) 7. These bounds remained≥ best known for over half a century, but recently≤ de Grey [4] proved χ(R2) 5 by providing a 5-chromatic unit-distance graph on 1581 vertices. A≥ smaller 5-chromatic unit-distance graph with only 553 vertices has been produced by Heule [9]. When a graph is not unit-distance, we call it forbidden. Unit-distance graphs can be frustrating to study in part because it is typically difficult to determine if a given graph is unit-distance or forbidden. For instance, it was conjectured by Chv´atal[2] that the so-called Heawood graph on 14 vertices was forbidden. Decades later, Gerbracht [7] refuted this conjecture by providing several unit-distance embeddings of the Heawood graph. More generally, Schaefer [11] has shown that deciding whether a given graph is unit-distance has the same complexity as deciding the truth of sentences in the existential theory of the real numbers; this is known to be NP-hard. To get a feeling for the problem, consider the two graphs on 9 vertices and 15 edges depicted in Figure1. We will show that one of these is unit-distance (see Figure5), while the other is forbidden (see Lemma 32). The goal of this article is to better understand unit-distance graphs by studying small obstructions. We say that a forbidden graph is minimal when each of its proper subgraphs is unit-distance. It is easy to see that both the complete graph K4 and the complete bipartite graph K2,3 are minimal forbidden graphs. In fact, every graph on up to 5 vertices is unit-distance if 108 unit-distance graphs in the plane and only if it does not contain either K4 or K2,3 as a subgraph. Chilakamarri and Mahoney [1] extended this observation by proving that a set of six graphs, which we denote by 7 and depict in Figure2, is the complete set of minimal forbidden graphsF≤ on up to 7 vertices. In other words, every graph on up to 7 vertices is forbidden if and only if it contains a subgraph isomorphic to an element of 7. Here we will extend their result to provide F≤ the complete list of minimal forbidden graphs on up to 9 vertices. Let 9 denote the set of 74 graphs depicted in Appendix A of the arXiv versionF≤ of this paper [8], which consists of 7, 13 graphs on 8 vertices, and 55 graphs on 9 vertices. We follow the notationF≤ of Chilakamarri and Mahoney and label these graphs F (n, m, i), where n indicates number of vertices, m indicates number of edges, and the last index i indicates only the order of appearance within this article. Our main result is: Theorem 1. A graph on at most 9 vertices is forbidden if and only if it contains a subgraph isomorphic to an element of 9. F≤ We will begin in the next section by introducing terminology and conventions that we will use throughout. In Section 3, we will prove Theorem9, which classifies the minimal forbidden graphs on 8 vertices. In Section 4, we will prove Theorem 33, which classifies the minimal forbidden graphs on 9 vertices. Theorem1 follows by combining the classification of 7 with Theorems9 and 33. The arXiv version of this paper [8] contains twoF≤ useful appendices. In Appendix A, we collect the set of minimal forbidden graphs on up to 9 vertices, and in Appendix B, we report coordinates for embedded unit-distance graphs that we use in the proofs of Theorems9 and 33. Our approach relies heavily on the freely available computational tools nauty [10] (to generate graphs) and SageMath [15] (to work with them). In the proof of Theorem 33, we found it necessary to use the implementation of cylindrical algebraic decomposition [3] within Mathematica [16] to generate embeddings for the two unit-distance graphs depicted in Figure5. We intend for our computations to be readily reproducible, and we have made our code available with the arXiv version of this paper [8]. F (4, 6, 1) F (5, 6, 1) F (6, 9, 1) F (7, 10, 1) F (7, 11, 1) F (7, 11, 2) Figure 2: Chilakamarri and Mahoney [1] proved that these six graphs form the set 7 of all minimal forbidden graphs on up to 7 vertices. F≤ 109 Globus and Parshall 2 Preliminaries We define an embedding of a unit-distance graph G on vertices V to be an injection ϕ: V R2 with the property that every pair of adjacent vertices v, w V satisfies→ ϕ(v) ϕ(w) = 1. We will say that G is rigid if for every pair∈ of its embeddings| −ϕ, ψ and| every pair of vertices v, w V , we have ϕ(v) ϕ(w) = ψ(v) ψ(w) . We will frequently fix coordinates∈ for rigid subgraphs| − of| a unit-distance| − graph| without impacting the generality of our arguments. We will say that a unit-distance graph is embedded when its vertices are distinct points in R2 and each of its edges are line segments of length 1. We allow for the possibility that non-adjacent vertices are distance 1 apart. Elementary geometry ensures that within any embedded unit-distance graph, every 3-cycle forms an equilateral triangle and every 4-cycle forms a rhombus. We will use these observations extensively in the following form. Lemma 2. For any embedded unit-distance graph G, the following hold. (i) The angle between any two edges of a 3-cycle in G is π/3. (ii) Opposite edges of a 4-cycle in G are parallel. We adopt the following conventions when drawing embedded unit-distance graphs. Black dots and solid line segments represent the vertices and edges, respectively, of the graph under consideration. White dots will represent relevant points of R2 that are determined by being distance 1 from other points pictured. We use dashed edges to represent pairs of points that are required to be distance 1 apart despite not necessarily corresponding to adjacent vertices in the graph under consideration. For instance, we will frequently consider the following unit-distance graph: x y Due to Lemma2, it must either be the case that x y = 1 or that x lies exactly distance 1 away from the other common unit-distance| − | neighbor of the 110 unit-distance graphs in the plane Figure 3: Each graph depicts a totally unfaithful unit-distance graph, drawn with solid edges, with non-adjacent vertices that are necessarily distance 1 apart in every embedding, represented with dashed edges. To see that each graph is totally unfaithful in this fashion, it is enough to apply Lemma2. two neighbors of y. We can separate these exhaustive cases by considering the two following classes of embeddings: x x y y Of course, neither embedding pictured is rigid, but we intend for these illustrations to represent the equivalence class of embeddings with the same constraints imposed by the solid and dashed unit-distance edges.